矢量导数

直角坐标系、球坐标系、柱坐标系中梯度、散度、旋度计算公式，备忘。

直角坐标系

$d\vec{l}=\hat i dx+\hat j dy+\hat k dz$

$dV=dxdydz$

$\nabla f=\hat i \frac{\partial f}{\partial x}+\hat j \frac{\partial f}{\partial y}+\hat k \frac{\partial f}{\partial z}$

$\nabla \cdot \vec{A}= \frac{\partial A_x}{\partial x}+ \frac{\partial A_y}{\partial y}+ \frac{\partial A_z}{\partial z}$

$\nabla \times \vec{A}= \hat i \left (\frac{\partial A_z}{\partial y}- \frac{\partial A_y}{\partial z}\right ) +\hat j \left (\frac{\partial A_x}{\partial z}- \frac{\partial A_z}{\partial x}\right ) +\hat k \left (\frac{\partial A_y}{\partial x}- \frac{\partial A_x}{\partial y}\right ) $

$\nabla^2 f= \frac{\partial^2 f}{\partial x^2}+ \frac{\partial^2 f}{\partial y^2}+ \frac{\partial^2 f}{\partial z^2}$

球坐标系

$d\vec{l}=\hat r dr+\hat {\theta} rd\theta+\hat {\phi} r\sin\theta d\phi$

$dV=rdrd\phi dz $

$\nabla f=\hat r \frac{\partial f}{\partial r}+\hat {\theta}\frac{1}{r} \frac{\partial f}{\partial \theta}+\hat {\phi} \frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi}$

$\nabla \cdot \vec{A}=\frac{1}{r^2} \frac{\partial}{\partial r}(r^2A_r)+ \frac{1}{r\sin\theta}\frac{\partial }{\partial \theta}(\sin\theta A_{\theta})+ \frac{1}{r\sin\theta}\frac{\partial }{\partial \phi} A_{\phi}$

$\nabla \times \vec{A}= \frac{\hat r}{r\sin\theta}\left[\frac{\partial }{\partial \theta}(\sin\theta A_{\phi})-\frac{\partial }{\partial \phi} A_{\theta} \right ]+ \frac{\hat {\theta}}{r}\left[\frac{1}{ \sin\theta}\frac{\partial }{\partial \phi} A_{r}-\frac{\partial }{\partial r} (rA_{\phi}) \right ]+\frac{\hat {\phi}}{r}\left[\frac{\partial }{\partial r} (rA_{\theta})-\frac{\partial }{\partial \theta} A_{r} \right ] $

$\nabla^2 f= \frac{1}{r^2}\frac{\partial }{\partial r} \left (r^2\frac{\partial f}{\partial r}\right)+\frac{1}{r^2\sin\theta }\frac{\partial }{\partial \theta} \left (\sin\theta\frac{\partial f}{\partial \theta}\right)+ \frac{1}{r^2\sin^2\theta }\frac{\partial^2 f}{\partial \phi^2}$

柱坐标系

$d\vec{l}=\hat r dr+\hat {\phi} r d\phi + \hat k dz$

$dV=r^2\sin\theta dr d\theta d\phi$

$\nabla f=\hat r \frac{\partial f}{\partial r}+\hat {\phi}\frac{1}{r} \frac{\partial f}{\partial \phi}+\hat k \frac{\partial f}{\partial z}$

$\nabla \cdot \vec{A}= \frac{1}{r}\frac{\partial }{\partial r}(rA_r)+ \frac{1}{r}\frac{\partial A_{\phi}}{\partial \phi}+ \frac{\partial A_z}{\partial z}$

$\nabla \times \vec{A}= \hat r \left (\frac{1}{r}\frac{\partial A_z}{\partial \phi}- \frac{\partial A_{\phi}}{\partial z}\right ) +\hat {\phi} \left (\frac{\partial A_r}{\partial z}- \frac{\partial A_z}{\partial r}\right ) +\hat k \left (\frac{1}{r}\frac{\partial }{\partial r}(rA_{\phi})- \frac{1}{r}\frac{\partial A_r}{\partial \phi} \right ) $

$\nabla^2 f= \frac{1}{r}\frac{\partial }{\partial r}\left (r\frac{\partial f}{\partial r}\right)+\frac{1}{r^2 }\frac{\partial^2 f}{\partial \phi^2}+ \frac{\partial^2 f}{\partial z^2}$